Theorem relcmpcmet 24842

Description: If 𝐷 is a metric space such that all the balls of some fixed size are relatively compact, then 𝐷 is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)

Hypotheses

Ref	Expression
relcmpcmet.1	⊢ 𝐽 = (MetOpen‘𝐷)
relcmpcmet.2	⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
relcmpcmet.3	⊢ (𝜑 → 𝑅 ∈ ℝ+)
relcmpcmet.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)

Assertion

Ref	Expression
relcmpcmet	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))

Distinct variable groups:   𝑥,𝐷   𝑥,𝐽   𝜑,𝑥   𝑥,𝑅   𝑥,𝑋

Proof of Theorem relcmpcmet

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcmpcmet.2	. 2 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
2		metxmet 23847	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
4	3	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝐷 ∈ (∞Met‘𝑋))
5		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (CauFil‘𝐷))
6		relcmpcmet.3	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ+)
7	6	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑅 ∈ ℝ+)
8		cfil3i 24793	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘𝐷) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
9	4, 5, 7, 8	syl3anc 1371	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
10	3	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐷 ∈ (∞Met‘𝑋))
11		relcmpcmet.1	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘𝐷)
12	11	mopntopon 23952	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
13	10, 12	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐽 ∈ (TopOn‘𝑋))
14		cfilfil 24791	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (Fil‘𝑋))
15	3, 14	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (Fil‘𝑋))
16	15	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑓 ∈ (Fil‘𝑋))
17		simprr 771	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
18		topontop 22422	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
19	13, 18	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐽 ∈ Top)
20		simprl 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑥 ∈ 𝑋)
21	6	rpxrd 13019	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℝ*)
22	21	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑅 ∈ ℝ*)
23		blssm 23931	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝑋)
24	10, 20, 22, 23	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝑋)
25		toponuni 22423	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
26	13, 25	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑋 = ∪ 𝐽)
27	24, 26	sseqtrd 4022	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ ∪ 𝐽)
28		eqid 2732	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
29	28	clsss3 22570	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ ∪ 𝐽)
30	19, 27, 29	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ ∪ 𝐽)
31	30, 26	sseqtrrd 4023	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋)
32	28	sscls 22567	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ∪ 𝐽) → (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
33	19, 27, 32	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
34		filss 23364	. . . . . . . 8 ⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ ((𝑥(ball‘𝐷)𝑅) ∈ 𝑓 ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓)
35	16, 17, 31, 33, 34	syl13anc 1372	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓)
36		fclsrest 23535	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) = ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
37	13, 16, 35, 36	syl3anc 1371	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) = ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
38		inss1 4228	. . . . . . 7 ⊢ ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ⊆ (𝐽 fClus 𝑓)
39		eqid 2732	. . . . . . . . 9 ⊢ dom dom 𝐷 = dom dom 𝐷
40	11, 39	cfilfcls 24798	. . . . . . . 8 ⊢ (𝑓 ∈ (CauFil‘𝐷) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
41	40	ad2antlr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
42	38, 41	sseqtrid 4034	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ⊆ (𝐽 fLim 𝑓))
43	37, 42	eqsstrd 4020	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ⊆ (𝐽 fLim 𝑓))
44		relcmpcmet.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
45	44	ad2ant2r 745	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
46		filfbas 23359	. . . . . . . . . 10 ⊢ (𝑓 ∈ (Fil‘𝑋) → 𝑓 ∈ (fBas‘𝑋))
47	16, 46	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑓 ∈ (fBas‘𝑋))
48		fbncp 23350	. . . . . . . . 9 ⊢ ((𝑓 ∈ (fBas‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓) → ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓)
49	47, 35, 48	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓)
50		trfil3 23399	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋) → ((𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ↔ ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓))
51	16, 31, 50	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ↔ ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓))
52	49, 51	mpbird 256	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
53		resttopon 22672	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
54	13, 31, 53	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
55		toponuni 22423	. . . . . . . . 9 ⊢ ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) = ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
56	54, 55	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) = ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
57	56	fveq2d 6895	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) = (Fil‘∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))))
58	52, 57	eleqtrd 2835	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))))
59		eqid 2732	. . . . . . 7 ⊢ ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) = ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
60	59	fclscmpi 23540	. . . . . 6 ⊢ (((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp ∧ (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅)
61	45, 58, 60	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅)
62		ssn0 4400	. . . . 5 ⊢ ((((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ⊆ (𝐽 fLim 𝑓) ∧ ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅) → (𝐽 fLim 𝑓) ≠ ∅)
63	43, 61, 62	syl2anc 584	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 fLim 𝑓) ≠ ∅)
64	9, 63	rexlimddv 3161	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝑓) ≠ ∅)
65	64	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)
66	11	iscmet 24808	. 2 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
67	1, 65, 66	sylanbrc 583	1 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ∖ cdif 3945   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322  ∪ cuni 4908  dom cdm 5676  ‘cfv 6543  (class class class)co 7411  ℝ^*cxr 11249  ℝ⁺crp 12976   ↾_t crest 17368  ∞Metcxmet 20935  Metcmet 20936  ballcbl 20937  fBascfbas 20938  MetOpencmopn 20940  Topctop 22402  TopOnctopon 22419  clsccl 22529  Compccmp 22897  Filcfil 23356   fLim cflim 23445   fClus cfcls 23447  CauFilccfil 24776  CMetccmet 24778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fi 9408  df-sup 9439  df-inf 9440  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-n0 12475  df-z 12561  df-uz 12825  df-q 12935  df-rp 12977  df-xneg 13094  df-xadd 13095  df-xmul 13096  df-ico 13332  df-rest 17370  df-topgen 17391  df-psmet 20942  df-xmet 20943  df-met 20944  df-bl 20945  df-mopn 20946  df-fbas 20947  df-fg 20948  df-top 22403  df-topon 22420  df-bases 22456  df-cld 22530  df-ntr 22531  df-cls 22532  df-nei 22609  df-cmp 22898  df-fil 23357  df-flim 23450  df-fcls 23452  df-cfil 24779  df-cmet 24781

This theorem is referenced by:  cmpcmet  24843  cncmet  24846